General RecommendationsProspective teachers need mathematics courses that develop understanding of the mathematics that they will teach.All courses should develop careful reasoning and teach both mathematical reasoning and mathematical common sense.There needs to be partnerships between school teachers and mathematics faculty.Teachers should have opportunities for professional development. Middle School mathematics should be taught by mathematics specialists.

Polyas QuotationsThe traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes a, he says b, he means c, but it should be d. Some of his sayings are handed down from generation to generation. Geometry is the science of correct reasoning on incorrect figures. My method to overcome a difficulty is to go round it. Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

Polyas QuotationsIn order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression. If there is a problem you can't solve, then there is an easier problem you can't solve: find it. A GREAT discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it. All the above are from How to Solve It (Princeton 1945).

Polyas QuotationsMathematics consists of proving the most obvious thing in the least obvious way. Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988). Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper. Quoted in D J Albers, G L Alexanderson and C Reid, Mathematical People (Boston 1985). When introduced at the wrong time or place, good logic may be the worst enemy of good teaching. The American Mathematical Monthly 100 (3). A mathematician who can only generalize is like a monkey who can only climb up a tree, and a mathematician who can only specialize is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalize and specialize. Quoted in D MacHale, Comic Sections (Dublin 1993)

Chapter 1 Problem Solving

(1.1): Exploring with PatternsAn Example: What is 3x4?Jack has 3 marbles. Jill has 4 marbles. They want to pick two marbles of different colors. How many different ways are there to pick two such marbles?Inductive Reasoning: Reasoning based on examining several data sets or patterns. Inductive reasoning leads to conjectures. Counter Examples: These are specific examples that prove a conjecture is wrong. Using logic to show that an idea is true, is deductive reasoning.There are three important mathematical ideas in this section. Arithmetic Sequences, Geometric Sequences, and sequences that are neither arithmetic nor geometric.